Determinant of this 5x5 matrix.

I tried this probably like 20 times (not exaggerating) and I keep making a mistake. I usually get -30 but the answer is 30. I even checked with software and it's 30. So, could someone please point out what I'm doing wrong?

Third line, second matrix: should be a -4 multiplying the 2 x 2 determinant, because the sign associated with the expansion corresponding to -1 has itself a -1, thus keeping whatever sign you had before.

I got it wrong again but I tried to see why I was wrong and I cannot see how the -1 gets an additional negative like: -(-1). I tried to explain myself on the new attached paper. If more explanation is needed, please tell me.

I got it wrong again but I tried to see why I was wrong and I cannot see how the -1 gets an additional negative like: -(-1). I tried to explain myself on the new attached paper. If more explanation is needed, please tell me.

first substract row 1 from row 2 and develop then by the first column:

$\displaystyle \left|\begin{array}{rrrrr}2&0&0&2&0\\0&0&3&-2&0\\0&2&0&0&-3\\0&0&0&-3&-1\\0&1&3&0&0\end{array}\right|=$ $\displaystyle 2\cdot\left|\begin{array}{rrrr}0&3&-2&0\\2&0&0&-3\\0&0&-3&-1\\1&3&0&0\end{array}\right|$ . Now substract twice the 4th row from the 2nd one

I got it wrong again but I tried to see why I was wrong and I cannot see how the -1 gets an additional negative like: -(-1). I tried to explain myself on the new attached paper. If more explanation is needed, please tell me.

The signs associated with the determinant: $\displaystyle \left|\begin{array}{ccc}2&0&-3\\0&0&-1\\1&3&0\end{array}\right|$